Oort's conjecture on direct sums of Newton polygons of curves

For i=1,2i=1,2, let gig_i be positive integers, let ξi\xi_i be symmetric Newton polygons of height 2gi2g_i, and let g=g1+g2g=g_1+g_2. Define ξ=ξ1ξ2\xi=\xi_1\oplus\xi_2 to be the symmetric Newton polygon of height 2g2g obtained by taking the union of the slopes of ξ1\xi_1 and ξ2\xi_2, with the multiplicity of each slope equal to the sum of its multiplicities in ξ1\xi_1 and ξ2\xi_2.

Oort's conjecture. If ξi\xi_i occurs on Mgi\mathcal{M}_{g_i} for i=1,2i=1,2, then ξ1ξ2\xi_1\oplus\xi_2 occurs on Mg1+g2\mathcal{M}_{g_1+g_2}.

This conjecture predicts that Newton polygons realized by curves can be combined through direct sum. The paper studies some cases of the conjecture, while the general assertion remains open.

Sources & referencesView supporting material

Primary source

Rachel Pries, “Some cases of Oort's conjecture about Newton polygons”, arXiv:2306.11080 (2024).

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